### Video Transcript

Find the instantaneous rate of

change of 𝑓 of 𝑥 is equal to the square root of 𝑥 at 𝑥 equals 𝑥 one, which

is greater than zero.

Remember, the instantaneous

rate of change of a function 𝑓 of 𝑥 at a point 𝑥 equals 𝑎 is found by taking

the limit as ℎ approaches zero of the average rate of change function. That’s the limit as ℎ

approaches zero of 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎 all over ℎ. In this case, we know that 𝑓

of 𝑥 is equal to the square root of 𝑥, and we want to find the instantaneous

rate of change at 𝑥 equals 𝑥 one. So, we’ll let 𝑎 be equal to 𝑥

one. Let’s substitute what we know

into our formula. We want to compute the limit as

ℎ approaches zero of 𝑓 of 𝑥 one plus ℎ minus 𝑓 of 𝑥 one all over ℎ. We need to find the limit as ℎ

approaches zero of the square root of 𝑥 one plus ℎ minus the square root of 𝑥

one all over ℎ.

Now, we can’t do this with

direct substitution. If we do, we end up dividing by

zero and we know that to be undefined. And so instead, we multiply the

numerator and denominator of the function by the conjugate of the numerator, by

the square root of 𝑥 plus one plus ℎ plus the square root of 𝑥 one. On the denominator, we simply

have ℎ times the square root of 𝑥 one plus ℎ plus the square root of 𝑥

one. Then on the numerator, we have

the square root of 𝑥 one plus ℎ times the square root of 𝑥 one plus ℎ, which

is simply 𝑥 one plus ℎ. Then, we multiply the square

root of 𝑥 one plus ℎ by the square root of 𝑥, and negative the square root of

𝑥 one times the square root of 𝑥 one plus ℎ. When we find their sum, we get

zero.

So, all that’s left to do is to

multiply negative the square root of 𝑥 one by the square root of 𝑥 one. And we simply get negative 𝑥

one. 𝑥 one minus 𝑥 one is

zero. And then, we divide through by

ℎ. And so, this becomes the limit

as ℎ approaches zero of one over the square root of 𝑥 one plus ℎ plus the

square root of 𝑥 one. And we can now evaluate this as

ℎ approaches zero. We’re left with one over the

square root of 𝑥 one plus the square root of 𝑥 one, which is one over two

times the square root of 𝑥 one. The instantaneous rate of

change function of 𝑓 of 𝑥 is equal to the square root of 𝑥 is therefore one

over two times the square root of 𝑥 one.